To graph the points [tex]$(0,6)$[/tex], [tex]$\left(\frac{\pi}{2}, 7\right)$[/tex], [tex]$(\pi, 8)$[/tex], [tex]$\left(\frac{3 \pi}{2}, 7\right)$[/tex], and [tex]$(2 \pi, 6)$[/tex] and write an equation to model these points, follow these steps:
### Step-by-Step Solution:
#### Plot the Points:
1. Plotting [tex]$(0,6)$[/tex]: Start by plotting the point [tex]$(0,6)$[/tex] on the graph.
2. Plotting [tex]$\left(\frac{\pi}{2}, 7\right)$[/tex]: Next, find [tex]$\frac{\pi}{2}$[/tex] on the x-axis and plot the point at a height of 7.
3. Plotting [tex]$(\pi, 8)$[/tex]: Similarly, plot the point [tex]$(\pi, 8)$[/tex] where [tex]$\pi$[/tex] is on the x-axis and the height is 8.
4. Plotting [tex]$\left(\frac{3 \pi}{2}, 7\right)$[/tex]: Move to [tex]$\frac{3 \pi}{2}$[/tex] on the x-axis and plot at a height of 7.
5. Plotting [tex]$(2 \pi, 6)$[/tex]: Finally, find [tex]$2\pi$[/tex] on the x-axis and plot the point at height 6.
#### Analyzing the Pattern:
- Notice that the points rise and fall in a symmetrical, periodic manner, suggesting a sinusoidal pattern.
- The points demonstrate an oscillation pattern typical of sine or cosine functions with a period of [tex]$2\pi$[/tex].
#### Determine the Equation:
- The pattern suggests using a sine or cosine function. In this case, we will use the cosine function as it starts at the middle value and goes up, then down, and back up.
- Amplitude (a):
- The maximum value is 8 and the minimum value is 6.
- Thus, the middle value (average) is [tex]$\frac{8+6}{2} = 7$[/tex].
- Amplitude [tex]$a$[/tex] is half the distance between the maximum and minimum value: [tex]\[ a = \frac{8 - 6}{2} = 1 \][/tex]
- Vertical Shift (k):
- The middle or average value that the function oscillates around is 7, so [tex]$k = 7$[/tex].
- Period:
- The period of a cosine function [tex]$f(x) = a\cos(bx) + k$[/tex] is determined by [tex]$2\pi/b$[/tex].
- Since the period is [tex]$2\pi$[/tex],
- we get [tex]$b = 1$[/tex] (because [tex]$2\pi/1 = 2\pi$[/tex]).
##### Equation:
Combining these values, the equation modeling the points is:
[tex]\[ y = 1 \cos(x) + 7 = \cos(x) + 7 \][/tex]
- The equation takes the form:
[tex]\[ y = a \cos(bx) + k \][/tex]
with [tex]\(a = 1\)[/tex] (amplitude), [tex]\( b = 1 \)[/tex] (period coefficient), and [tex]\( k = 7 \)[/tex] (vertical shift).
### Conclusion:
To model the given points graphically, you will have:
[tex]\[ y = \cos(x) + 7 \][/tex]
This equation fits the observations, with the points oscillating between [tex]$8$[/tex] and [tex]$6$[/tex], centered around [tex]$7$[/tex].
